Holders La Rochelle reach Champions Cup quarters after late Libbok miss
The hosts had led 13-0 at the break before an impressive comeback from the French side, who had a 22-16 advantage before Libbok failed with his kick in windy conditions following Suleiman Hartzenberg's try.
La Rochelle, bidding for a third successive Champions Cup title, had lost to the same opponents in December's pool stage meeting when a successful late Libbok conversion proved decisive.
"We knew the dangers of them scoring, we backed our defence but if by chance they scored the wind was horrible," La Rochelle lock Ultan Dillane told reporters.
"It was a nightmare seeing it (Libbok's successful kick) around Christmas time but fate was on our side today."
La Rochelle will face the winners of the game between four-time winners Leinster and Leicester, who play later on Saturday, in the quarter-finals next weekend.
Ronan O'Gara's side were handed a pre-match boost with giant Australia lock Will Skelton returning after two months out with injury for one of five repeat fixtures from the group stage in the last 16.
The Stormers had four Rugby World Cup winners in their matchday squad, including fly-half Libbok.
The hosts led 13-0 before the half-hour mark as Libbok slotted eight points and scrum-half Herschel Jantjies crossed for a try.
'Patient' Bordeaux-Begles
The Stormers stretched their lead to 16-0 with a third Libbok penalty, but then lost a third back-rower to injury as Hacjivah Dayiman followed Deon Fourie and Ben-Jason Dixon off the field.
Dayiman's departure sparked La Rochelle's comeback as prop Louis Penverne and No 8 Gregory Alldritt crossed either side of a penalty from fly-half Antoine Hastoy to put La Rochelle 17-16 up with 18 minutes to play.
La Rochelle extended their advantage with a 68th-minute touchdown from Argentina prop Joel Sclavi but Hastoy missed the conversion, setting up a tense finish with the Stormers within a converted try from a quarter-final spot.
Hartzenberg crossed with 35 seconds to play from a delicate Libbok cross-kick, setting the latter up for a match-winning conversion with the clock in the red.
Libbok stepped up, with Willemse holding the ball on the tee due to gusty conditions, but his effort drifted wide as O'Gara's champions survived.
Elsewhere, Exeter downed English rivals Bath 21-15 and will meet either Toulouse or Racing 92 in the next round, with the French clubs meeting on Sunday.
Bordeaux-Begles saw off Saracens 45-12, having thumped the three-time former champions 55-15 in the pool stage, thanks to doubles from France winger Louis Bielle-Biarrey, centre Nicolas Depoortere and fly-half Mateo Garcia.
The French club will host Harlequins in the last eight after the English outfit beat Glasgow 28-24 on Friday.
"We had to be patient and trust our systems and if we held the ball, build pressure, we were sure it would give at one point," Bordeaux-Begles's prop Ben Tameifuna told TNT Sports.
"We're one step closer to what we want to achieve as a group."
There is also repeat on Sunday of the 2000 final as Munster head to English Premiership leaders Northampton for a chance to meet the Bulls, who thrashed a second-string Lyon 59-19.